group the 29 topologies of 123 into homeomorphic classesSolu

group the 29 topologies of {1,2,3} into homeomorphic classes.

Solution

nine distinct classes exists where the following lists out one element from each class

for example if you take the second element {, {3}, {1,2,3}} then in this class we have the other elemnts as {, {1}, {1,2,3}}, {, {2}, {1,2,3}}

and in third class we have {, {1,2}, {1,2,3}}, {, {2,3}, {1,2,3}}, {, {1,3}, {1,2,3}

and so on. you can just list them out strucure wise .

first class and last class have one and only one element which is listed.